Università degli Studi di Firenze, 18-19 April 2012 WAVE ENERGY UTILIZATION António F. O. Falcão Instituto Superior Técnico, Universidade Técnica de Lisboa INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
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Università degli Studi di Firenze, 18-19 April 2012
WAVE ENERGY UTILIZATION
António F. O. Falcão
Instituto Superior Técnico,
Universidade Técnica de Lisboa
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
WAVE ENERGY
SOLAR
ENERGY
WIND
ENERGY
WAVE
ENERGY
Typical values of wave energy flux (annual average):
Deep water: 6-70 kW/m
Near shore: lower values,
Depending on:
• bottom slope
• local depth (wave breaking)
• bottom roughness (friction)
• bottom configuration (diffraction, refraction)
Close to the surface (h<20m):
density flux of energy (kW/m2) much
higher than wind energy
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
World distribution of wave energy level Annual-averaged values in kW/m (deep water, open sea)
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
THE WAVES AS ENERGY RESOURCE
The waves are generated by the wind.
In deep water ( > 100 - 200m ) they travel large distances
(thousands of km) practically without dissipation.
The characteristics of the waves (height, period, etc.) depend
on:
Sea surface area acted upon by the wind: “fetch”
Duration of wind action
“Swell”: wave generated at a long distance (mid ocean).
“Wind sea”: waves generated locally.
In general, swell is more energetic than wind sea.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
The free-surface is unknown, which makes the problem non-linear.
FLUID MOTION IN WAVES
• Perfect fluid (no viscosity)
Incompressible flow
Irrotational flow
0 V
VV or 0
02
Boundary conditions
• At the free-surface:
At the bottom:
atpp
0 nV
In general the boundary condition is applied at the undisturbed free-surface
(flat surface): LINEAR THEORY.
Laplace equation
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
z
x crest
trough
hz
0
The simplest solution: the sinusoidal regular wave
0
22sin)(
2coshconst
xt
Thz
0
22sin
2expconst
xt
Tzh If
T = period (s), f = 1/T = frequency (Hz or c/s),
= radian frequency (rad/s),
λ = wavelength (m), = wave number (m-1)
Tf 22
/2k
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
)sin(22
sin),( 00
kxtAxt
TAtx
A = wave amplitude
H = 2A = wave height (from trough to crest)
The disturbance decreases with the distance to the surface.
In deep water, the decrease is exponential: the disturbance practically
vanishes at a depth of about 1/2 wavelength.
Free-surface elevation
0
22sin
2expconst
xt
Tz
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
In deep water, the water particles have circular orbits.
The orbit radius decreases exponentially with the distance to the
surface.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
In water of finite depth, the orbits are ellipses.
The ellipses become flat near the bottom.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Propagation velocity (phase velocity)
kTc
c
h
gc
tanh
From the boundary condition at the sea surface:
The velocity of propagation c depends on the wave period T (or
frequency ω or f ) and also on the water depth h.
The sea is a dispersive medium for surface waves.
The speed of sound in air is independent of frequency.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
c
c
h
g
h2
h
INTERNATIONAL PHD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Limiting situations
In shallow water (in practice if h << λ) khkh )tanh(
ghc
c does not depend on T
c
h
gc
tanh
In deep water (in practice if h > λ/2) :
2
gTg
k
gc
1)tanh(tanh khc
h
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Example s8T2
m/s 8,9g
m/s 5,122
88,9
2
gTcDeep water m 10085,12 cT
Intermediate water depth h = 15 m rad/s 785,08
22
T
c
h
gc
tanh
cc
15785,0tanh
8,9
785,0 m/s 2,10c
m 8,8182,10 cT
Shallow water h = 1 m
m/s 0,2581,3 cTm/s 1,318,9 ghc
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
1 3,10 24,8
3 5,25 42,0
5 6,63 53,0
10 8,86 70,9
15 10,22 81,8
20 11,09 88,7
25 11,65 93,2
30 12,00 96,0
40 12,33 98,6
50 12,44 99,5
12,48 99,8
(m) h (m/s) c (m)
s8T
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
wave crests
Refraction effects due to bottom bathymetry
The propagation velocity c decreases with decreasing depth h.
As the waves propagate in decreasing depth, their crests tend to
become parallel to the shoreline
shoreline
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
rays crests
shoreline
shoreline
Dispersion of energy at a bay.
Concentration of energy at a headland.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Group velocity or velocity of propagation of energy
The velocity of propagation of wave energy, , is different from (smaller than)
the phase velocity or velocity of propagagtion of the crests c. gc
In deep water, it is cc g2
1
In sound waves, there is no difference between the two velocities.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
ENERGY OF THE WAVES
Kinetic energy (circular or elliptic orbits)
Potential energy (sea surface is not plane)
v
In deep water, energy per unit horizontal area, time-averaged:
)2( AH 22
potkin16
1
4
1gHgAWW
)mJ(222
potkin8
1
2
1gHgAWWW
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
We are more interested in the energy flux across
a vertical plane parallel to the wave crests (from
bottom to surface).
Energy flux per unit length along wave crests
(time-averaged)
In deep water (W/m)
Note:
The energy flux is proportional to the wave period T and to the
square of the wave amplitude A (or the wave height H = 2A).
This is energy flux from surface to bottom.
Most of the contribution to E is from the upper layer close to the
sea surface.
TAgE22
8
1
1
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
REAL IRREGULAR WAVES
Real waves are not sinusoidal.
However, they can be represented with good approximation as
superpositions of sinusoidal (regular) waves.
If , we have a continous spectrum. N
Frequently a power spectum is defined (rather than for
amplitude).
)(S
N
n
nnnn xktAtx
1
)sin(),( surface elevation
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
)sm(
)(
2
S
(rad/s)
Example of power spectrum
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
In practice, for numerical simulations, the spectrum has to
be discretized
max
min
sin),(
N
Nnnnnn xktAtx
0 nn
nk
0
)(4 00 nSA n
)20( nn
corresponding wave number
small frequency interval
random phase
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Power spectrum
For a given sea state, the power spectrum my be obtained from
records of wave measurements (surface elevation) and the
application of spectral analysis.
In numerical simulations, spectral distributions are used that fit large
classes of sea states.
One is the Pierson-Moskowitz spectral distribution:
sH
eT
= significant wave height
= energy period
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
s10eT
m2sH
)(S
)1054exp(263)(44542
ees TTHS
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
The moments of the wave spectrum
0
)( dffSfmn
n
2f
(rad/s)
04 mH s Significant wave height
31HH s = mean value of the highest 1/3 of wave heights
0
1
m
mT e
Energy period
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
)(S peak energy
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
100 150 200 250 300
t s
1
0.5
0
0.5
1
1.5
m
Example
Simulated time-series of surface elevation at a given point from a Pierson-
Moskowitz spectrum discretized into 225 sinusoidal harmonics
s10eT m2sH
200
t (s)
)( t
(m)
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Energy flux of irregular waves
If the spectral distribution is known, the energy flux may be obtained
as the sumation of the energy fluxes of the sinusoidal harmonics.
For a Pierson-Moskowitz spectrum, it is (in deep water)
(kW/m)
)kW/m(E energy flux por unit wave-crest length
energy period )s(eT
)m(sH significant wave height
s10eT
E = 44.1 kW/m m3sH
249.0 se HTE
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Directional spread of the waves
As real waves are not generated at a single point on the ocean, their
direction is not well defined: there is a directional spread.
This applies to a sea state or to a (annual-averaged) wave climate.
A two-dimensional spectrum may be defined :
),( S
)(cos 0
2
sCosine law is frequently used:
Larger exponent s means more concentrated directional spectrum.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave and Resource Statistics The wave climate may be regarded as a set of sea states, each sea state (i,j)
characterized by
- Significant wave height
- Mean energy period
- Frequency of occurrence
isH ,
jeT ,
jiF ,
1
,
, ji
jiF
Scatter diagram
jiF ,
jeT ,
isH , <4s 4-5m 5-6s 6-7s 7-8s ...
<0.5 m
0.5-1m
1-1.5m
1.5-2m
2-2.5m
2.5-3m
....
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave and Resource Statistics
Off West Portugal, h = 100m, all directions
Maximum frequency of occurrence -
Annual Relative Frequency in terms of (Hs,Te)
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave and Resource Statistics
Maximum energy contribution -
Off West Portugal, h = 100m, all directions
Annual Energy Distribution in terms of (Hs ,Te)
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave and Resource Statistics
1 2 3 4 5 6 7 8 9 10 11 12
Months
0
1
2
3
4
5
6
7
Hs(m
)
Mean
Quantiles 7.5%-92.5%
Quantiles 2.5%-97.5%
Hs monthly variation- 39ºN - Lisbon
From: WERATLAS
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave and Resource Statistics
1 2 3 4 5 6 7 8 9 10 11 12
Months
0
50
100
150
200
250
300
P(k
W/m
)
Mean
Quantiles 7.5%-92.5%
Quantiles 2.5%-97.5%
Power Monthly Variation
From: WERATLAS
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Units:
kW/m
Source: WERATLAS, 1996
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Units:
kW/m
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Units:
kW/m
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Units:
kW/m
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Units:
kW/m
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave Energy Resource
From: Barstow,
Mollison & Cruz
Seasonal Variation
Seasonal variations are much larger in the Northern Hemisphere than in the
Southern Hemisphere (an important advantage)
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Wave Energy Resource
Seasonal Variation
SOUTH
NORTH
From: Barstow, Mollison & Cruz.
Lowest mean monthly wave power relative to annual mean
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
2m 2 x 40 = 80 kW = 108 CV
How much wave power along the Portuguese coast?
250 000
Annual average 40 kW/m
500 km
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Theoretical Resource - A top level statement of the energy contained in the
entire resource
Technical Resource – Part that can be exploited based on existing
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
Similarities and contrasts between the wind
energy resource and the wave energy resource
Over time-scales of a few wave periods, the waves are random, like wind
turbulence.
Due to the own nature of waves, the absorbable power is highly oscillating
and practically discontinuous.
Waves result from the integrated action of the wind over large ocean areas
(thousands of square km) and several hours or days their variability
is less than for wind, and they are more predictable
Comparison between time-averages (over tens of minutes to one hour):
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
<200m
several
km
WIND
The wind velocity profile extends over
several km.
A wind farm explores a tiny sublayer
Most of the wave energy flux is
concentrated near the surface
A wave farm can absorb a large part of
the wave energy flux.
Typically, the energy flux per unit vertical area for waves
near the surface is about 5 times larger than for wind.
Waves are a more concentrated form of energy than
wind.
20m
WAVES
Most of the wave energy flux is
concentrated near the surface
A wave farm can absorb a large part of
the wave energy flux.
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012
END OF PART 1. WAVE
ENERGY RESOURCE
INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012